When Euler solved the Basel Problem in 1735, his proof involved the fact that $\frac{\sin x}x$ is the product of the linear factors given by the roots, i.e.
$$\frac{\sin x}x=\cdots\left(1+\frac x{2\pi}\right)\left(x+\frac x\pi\right)\left(1-\frac x\pi\right)\left(1-\frac x{2\pi}\right)\cdots$$
However, one could argue that $f(x)\frac{\sin x}x=\cdots\left(1+\frac x{2\pi}\right)\left(x+\frac x\pi\right)\left(1-\frac x\pi\right)\left(1-\frac x{2\pi}\right)\cdots$ for any $f(x)$ which does not have any roots and is finite for all $x\in\mathbb{R}$, since the roots of $f(x)\frac{\sin x}x$ and $\frac{\sin x}x$ are the same. An example would be $e^x\frac{\sin x}x$.
So, why is it valid to say that $\frac{\sin x}x=\cdots\left(1+\frac x{2\pi}\right)\left(x+\frac x\pi\right)\left(1-\frac x\pi\right)\left(1-\frac x{2\pi}\right)\cdots$?
